[Paper Review] Bayesian Modeling of Intersectional Fairness: The Variance of Bias

James R. Foulds, Rashidul Islam+2 more|arXiv (Cornell University)|Nov 18, 2018

Gender Politics and Representation4 citations

TL;DR

This paper proposes a Bayesian probabilistic framework to reliably estimate intersectional fairness in machine learning, addressing data sparsity across multiple protected attributes. By employing hierarchical Bayesian modeling and variational inference, it provides robust uncertainty estimates for fairness metrics—demonstrating superior performance over empirical estimates, especially in sparse regimes like the COMPAS and census datasets.

ABSTRACT

Intersectionality is a framework that analyzes how interlocking systems of power and oppression affect individuals along overlapping dimensions including race, gender, sexual orientation, class, and disability. Intersectionality theory therefore implies it is important that fairness in artificial intelligence systems be protected with regard to multi-dimensional protected attributes. However, the measurement of fairness becomes statistically challenging in the multi-dimensional setting due to data sparsity, which increases rapidly in the number of dimensions, and in the values per dimension. We present a Bayesian probabilistic modeling approach for the reliable, data-efficient estimation of fairness with multi-dimensional protected attributes, which we apply to two existing intersectional fairness metrics. Experimental results on census data and the COMPAS criminal justice recidivism dataset demonstrate the utility of our methodology, and show that Bayesian methods are valuable for the modeling and measurement of fairness in an intersectional context.

Motivation & Objective

• Address the challenge of data sparsity in measuring fairness across multiple, intersecting protected attributes such as race, gender, and nationality.
• Develop a statistically sound, data-efficient method to estimate fairness metrics with quantified uncertainty in intersectional AI fairness.
• Overcome limitations of empirical probability estimates in small-sample intersectional groups, where traditional methods fail due to unreliable statistics.
• Provide a practical, scalable framework for auditing existing AI systems for intersectional discrimination, especially in high-stakes domains like criminal justice.
• Demonstrate the utility of Bayesian hierarchical models and model averaging for improving reliability in fairness measurement under data scarcity.

Proposed method

• Propose a Bayesian probabilistic framework that models fairness metrics as random variables with posterior distributions, enabling uncertainty quantification.
• Implement four statistical models—empirical Bayes (EB), full Bayes (FB), logistic regression (LR), and deep neural networks (DNN)—with a novel hierarchical Bayesian logistic regression (HLR) model for improved stability.
• Use variational inference to approximate the posterior distributions of model parameters, enabling scalable computation on large, sparse datasets.
• Apply Bayesian model averaging (BMA) to combine predictions across models, enhancing robustness and reducing variance in fairness estimates.
• Leverage posterior predictive distributions to generate uncertainty-aware fairness estimates, including credible intervals for metrics like differential fairness (DF) and subgroup fairness (SF).
• Integrate bootstrap resampling with Bayesian inference to further improve estimation reliability in small-sample regimes.

Experimental results

Research questions

• RQ1 How can fairness be reliably measured in the presence of data sparsity across multiple protected attributes in intersectional AI fairness?
• RQ2 What Bayesian modeling approach provides the most stable and accurate estimation of fairness metrics when intersectional groups are small or underrepresented?
• RQ3 To what extent does incorporating uncertainty via Bayesian inference improve fairness measurement compared to empirical probability estimates?
• RQ4 How do different Bayesian models (e.g., HLR, DNN, LR) compare in performance on real-world datasets like COMPAS and the UCI Adult census dataset?
• RQ5 Can Bayesian model averaging enhance the robustness and reliability of intersectional fairness estimation in sparse data regimes?

Key findings

• The hierarchical Bayesian logistic regression (HLR-FB) model consistently outperformed other models in estimating fairness metrics on sparse data, achieving the lowest L1 deviation from full-dataset ground truth.
• On the 1% subsample of the COMPAS dataset, HLR-FB achieved an L1 deviation of 0.393 for γ-SF and 0.015 for ε-DF, indicating high accuracy and low uncertainty.
• All Bayesian models, especially FB variants, showed significantly lower L1 deviations than empirical probability (PE) estimates, demonstrating improved reliability in sparse settings.
• The HLR-FB model predicted that the COMPAS system amplifies bias by approximately 0.5 in ε-DF, with a 95% credible interval between 0.35 and 0.65, indicating strong evidence of bias amplification.
• Posterior predictive distributions placed high probability on values exceeding the legal threshold of 0.2231 for the 80% rule, confirming substantial evidence of disparate impact in both true recidivism and COMPAS predictions.
• Bayesian model averaging improved stability but did not surpass the HLR-FB model in performance, suggesting HLR-FB is a strong standalone choice for intersectional fairness estimation.